Particles in a radioactive piece of material are decaying, and it is known that the number of decayed particles during a time period t (sec) is $Po(\lambda t)$-distributed. It is also known that the probability of there being at least one particle that has decayed during 60 seconds is 0.5. What value for $\lambda$ is the appropriate one to use in this model for radio active decay?
I am stumped on how to approach this problem. In particular I do not know how to make use of the given probability - P(at least one particle has decayed during a 60 second period)=0.5.
\begin{align} &\begin{cases} P(\text{at least one particle has decayed during a $60$ second}) = 0.5 \\ P(\text{at least one particle has decayed during a $60$ second}) = 1- P(\text{$0$ particle has decayed during a $60$ second}) \end{cases} \\ \Rightarrow &P(\text{$0$ particle has decayed during a $60$ second})= 1-0.5 = 0.5 \quad (1) \end{align}
By using $(1)$, we have \begin{align} &P(k \text{ events in 60 second}) = \frac{e^{-\lambda}\times \lambda^{k}}{k!} \Rightarrow \\ &P(0 \text{ events in 60 second}) = \frac{e^{-\lambda}\times \lambda^{0}}{0!} = 0.5 \Rightarrow e^{-\lambda} = 0.5 \Rightarrow \lambda = - ln(0.5) \end{align}